(p, q) forms. The Fr¨olicher Spectral Sequence ofX is the spectral sequence derived from the double complex (Ap,q(X), ∂, ∂) and (Enp,q, dn)⇒H∗(X,C).
It is well known that it degenerates at theE1 term ifXis a K¨ahler manifold and hence a non degenerating Fr¨olicher spectral sequence measures in some sense how far a manifold is from being a K¨ahler manifold.
The historically first example for this phenomenon was the Iwasawa man-ifold for which E1 ≇E2 ∼=E∞. In the book of Griffith and Harris [GH78]
is repeated the question if there were examples with nonzero differential in
higher termsEnand in [CFUG99] several complex 3 dimensional cases were analysed which collapsed only at the E3 term.
The aim of the section is to describe a family of examples which show that principal holomorphic torus bundles and hence nilmanifolds can be arbitrarily far from being K¨ahler manifolds in the sense that the Fr¨olicher spectral sequence can be arbitrarily non degenerate. This is already true for 2-step nilpotent Lie algebras:
Example 1.17— We give for n at least 2 a Lie algebra gn with complex structure defined by the following structure equations. Let
{ω1, . . . , ωn, λ1, . . . , λn, η1. . . , ηn−1}
be a basis for g∗n1,0 := g∗n1,0C , the (1,0) part of the dual of gn and consider the differential d:g∗nC→Λ2g∗nC given by
dω1= ¯λ1∧η1
dωk=λ1∧η¯k−1+ (−1)nλ1∧λk k= 2, . . . , n dηk=dλk= 0 for all k
and by the complex conjugate equations. Obviously d2 = 0 and hence the Jacobi identity holds for the associated real Lie algebra structure. The Lie algebra gn is 2-step nilpotent and the centre, given by the subspace annihilated by theηk’s andλk’s and their complex conjugates, is a complex subspace with respect to this complex structure.
Let Gn be the associated simply-connected nilpotent Lie group and let An be the abelian subgroup corresponding to the centre of gn. Since the structure constants are integers we can choose a lattice Γ ⊂ Gn such that the nilmanifold Mn= Γ\Gn is a principal torus bundle over a torus.
The differential on the exterior algebra Λ∗g∗C induced by d can be de-composed in the usual way as d = ∂ +∂. Considering elements in g∗C as left-invariants differential forms on M we get an inclusion of differential bigraded algebras
Λ∗,∗g∗C֒→ A∗,∗(Mn).
It was proved in [CFUG99] that this inclusion induces for principal holo-morphic torus bundles an isomorphism of spectral sequences after the E1
term.
Proposition 1.18 — The Fr¨olicher spectral sequence of Mn does not de-generate at the En term. More precisely we have
dn([¯ω1∧η¯2∧ · · · ∧η¯n−1]) = [λ1∧ · · · ∧λn]6= 0, i.e., the mapdn:En0,n−1→Enn,0 is not trivial.
Proof. TheE0 term of the first quadrant spectral sequence associated to the double complex (Λ•,•g∗nC, ∂, ∂) is given byE0p,q= Λpg∗n1,0⊗Λqg∗n0,1 and d0 =∂. The differentiald1 is induced by∂.
Following the exposition in [BT82] (§14, p.161ff) we say that an element ofb0 ∈E0p,q lives to Er if it represents a cohomology class in Er or equiva-lently if it is a cocycle in E0, E1, . . . , Er−1. This is shown to be equivalent to the existence of a zig-zag of length r, that is, a collection of elements b1, . . . , br−1 such that
bi ∈E0p+i,q−i ∂b0= 0 ∂bi−1 =∂bi(i= 1, . . . r−1).
These can be represented as 0
b_0
∂
OO ∂ //
b_1
OO //
. ..
br−1_
OO //dn([b0]) = [∂br−1].
In this picture we have the first quadrant double complex given by (E0p,q, ∂, ∂) in mind in which this zig-zag lives.
Furthermoredn([b0]) = [∂br−1] is zero inErp+n,q−n+1 if and only if there exists an elementbr ∈Ep+n,q−n0 such that ∂br =∂br−1, i.e. we can extend the zigzag by one element.
We will now show thatb0 := ¯ω1∧η¯2∧ · · · ∧η¯n−1 admits a zigzag of length nwhich cannot be extended. Since ∂¯ω1 = 0 we have ∂b0= 0 and calculate
∂b0 =λ1∧η¯1∧ · · · ∧η¯n−1. Let us define:
b1 :=ω2∧η¯2∧ · · · ∧η¯n−1
bi :=ωi+1∧η¯i+1∧ · · · ∧η¯n−1∧λ2∧ · · · ∧λi (i= 2, . . . , n−2) bn−1 :=ωn∧λ2∧ · · · ∧λn−1
Then ∂bi−1 = ∂bi(i = 1, . . . n−1) and we have the desired zigzag. We conclude by saying that dn([b0]) = [∂bn−1] = [λ1 ∧. . . λn] 6= 0 because E0n,−1= 0 and the zig-zag cannot be extended.
Remark 1.19—Note that exchanging everyλi byηi in the above defini-tion ofgnwould yield an example of dimension 2nwith the same properties;
but we found it graphically more convincing to replace one of the ¯ηi’s by one of the λi’s in every step of the zigzag in the proof. This yields
Corollary 1.20— Forn≥2there exist2n-dimensional 2-step nilmanifolds with left-invariant complex structure, which are principal holomorphic torus bundles over a base of dimension n with fibre of dimension n, such that the Fr¨olicher spectral sequence does not degenerate at the En term.
2 Lie algebra Dolbeault cohomology
The aim of this section is to set up some Dolbeault cohomology theory for modules over Lie algebras with complex structure and prove Serre duality in this context. Our main application is the calculation of the cohomology groups of the tangent sheaf for nilmanifolds with a left-invariant complex structure in Section 3 but perhaps the notions introduced are of independent interest.
In the first subsection we define the notion of (anti-)integrable module and derive the elementary properties which are interpreted in geometric terms in section two. The third section is again devoted to algebra when we set up our machinery of Lie algebra Dolbeault cohomology while the fourth one will explain the geometric implications of our theory.
2.1 Integrable representations and modules
For the whole section let (g, J) be a Lie algebra with complex structure.
Several times we will refer to the Nijenhuis tensor (1) which was defined in Section 1.
A leftg-module structure on a vector spaceE is given by a bilinear map g×E →E (x, v)7→x·v
such that [x, y]·v=x·y·v−y·x·v. Note that this induces a mapg→EndE, a representation of g on E. If we want to stress the Lie algebra structure on EndE (induced by the structure of associative algebra by setting [a, b] = ab−ba) we use the notation gl(E). A representation or a left module structure correspond hence to a Lie algebra homomorphismg→gl(E).
In the sequel we want to combine these notions with complex structures both ong andE.
Let (g, J) be a Lie algebra with (integrable) complex structure andE a real vector space with (almost) complex structureJ′.
Definition 2.1— A representation ρ: (g, J)→EndE of gon E is said to be integrable if for all x∈g the endomorphism of E given by
N(x) := [J′,(ρ◦J)(x)] +J′[ρ(x), J′]
vanishes identically. In this case we say that(E, J′)is anintegrable (g, J )-module. We say that (E, J′) is antiintegrable if (E,−J′), the complex conjugated module, is an integrable g-module. A homomorphism of (anti-) integrable g-modules is a homomorphism of underlying g-modules that is C linear with respect to the complex structures.
The definition is motivated by the fact that the adjoint representation ad:g →Endg given by x7→ [x,−] is integrable in this sense if and only if the Nijenhuis tensor (1) vanishes, i.e., if (g, J) is a Lie algebra with complex structure. This is a special case of the next result.
Proposition 2.2— Let (E, J′) be vector space with complex structure and ρ:g→EndE a representation. Then the following are equivalent:
(i) ρ is integrable.
(ii) For allX ∈g1,0 the map ρ(X) has no component in Hom(E1,0, E0,1).
(iii) E1,0 is an invariant subspace under the action of g1,0.
(iv) ρ|g1,0 induces a complex linear representation onE1,0 by restriction.
Proof. The restriction ofρ to g1,0 is Clinear by definition since it is the complexification of the real representation restricted to a complex subspace.
Therefore condition (ii) is equivalent to (iii) and (iv).
It remains to prove (i)⇔ (ii).
LetX ∈g1,0 andV ∈V1,0. UsingJX =iX andJ′V =iV we calculate N(X)V = ([J′,(ρ◦J)(X)] +J′[ρ(X), J′])(V)
= (iJ′ρ(X)−iρ(X)J′+J′ρ(X)J′−J′2ρ(X))(V)
= 2iJ′ρ(X)V + 2ρ(X)V and see that
N(X)V = 0⇔J′ρ(X)V =iρ(X)V ⇔ρ(X)V ∈V1,0.
This proves (i)⇒(ii). Vice versa assume that (ii) holds. We decompose the elements in EC respectively in gC into their (1,0) and (0,1) parts. By the above calculation and its complex conjugate (the representation and hence the bracket are real and commute with complex conjugation) it remains to consider the mixed case. We have for all X, V as above
N(X) ¯V = (iJ′ρ(X)−iρ(X)J′+J′ρ(X)J′−J′2ρ(X))( ¯V)
=iJ′ρ(X) ¯V −ρ(X) ¯V −iJ′ρ(X) ¯V +−ρ(X) ¯V
= 0
and henceρ is integrable.
Corollary 2.3 — Let (E, J′) be vector space with complex structure and ρ:g→EndE a representation. Then the following are equivalent:
(i) ρ is antiintegrable.
(ii) For allX¯ ∈g0,1 the map ρ( ¯X) has no component in Hom(E1,0, E0,1).
(iii) E1,0 is an invariant subspace under the action of g0,1.
(iv) ρ|g0,1 induces a complex linear representation onE1,0 by restriction.
Proof. Exchange J′ by −J′ in the above proof.
Proposition 2.4 — Let ρ be an integrable representation on (E, J′). The bilinear mapδ given by
δ:g0,1×E1,0 →ρ EC→pr E1,0 ( ¯X, V)7→(ρ( ¯X)V)1,0 induces a complex linear representation of g0,1 on E1,0.
Proof. The map is clearly complex bilinear and it remains to prove the compatibility with the bracket. Let ¯X,Y¯ ∈g0,1 and V ∈E1,0 be arbitrary.
Note thatρ( ¯Y)V =δ( ¯Y , V) + (ρ( ¯Y)V)0,1. Then δ([ ¯X,Y¯], V)
=(ρ([ ¯X,Y¯])(V))1,0
= ρ( ¯X)ρ( ¯Y)V −ρ( ¯Y)ρ( ¯X)V1,0
=
ρ( ¯X)(δ( ¯Y , V) + (ρ( ¯Y)V)0,1)−ρ( ¯Y)(δ( ¯X, V) + (ρ( ¯X)V)0,1)1,0
= ρ( ¯X)δ( ¯Y , V)−ρ( ¯Y)δ( ¯X, V)1,0
+ (ρ( ¯X)(ρ( ¯Y)V)0,1−ρ( ¯Y)(ρ( ¯X)V)0,1
| {z }
of type (0,1)
)1,0
=δ( ¯X, δ( ¯Y , V))−δ( ¯Y , δ( ¯X, V)).
Here we used that the action ofg0,1 mapsE0,1 toE0,1 which is the complex conjugate of Proposition 2.2 (iii). Henceδ induces a g0,1-module structure
onE1,0 as claimed.
Lemma 2.5 — Let (E, J′) be an integrable (g, J)-module. Then the dual module with the induced g-module structure is antiintegrable.
Proof. Ifx ∈g and φ∈E∗ then the induced module structure is given by (x·φ)(v) =−φ(xv) for v ∈E. We have to show that for ¯X ∈g0,1 and Φ∈E∗1,0 the map ( ¯X·Φ) annihilatesE0,1. But if ¯V is in E0,1 then by the above proposition ¯XV¯ ∈E0,1 and Φ( ¯XV¯) = 0.
Remark 2.6— The above result seems unnatural only at first sight. If we considerE as left gl(E)-module in the canonical way then the complex structureJ ∈EndE acts on the left. The dual vector spaceE∗ comes with a natural action ofgl(E) on the right:
φ·A:=φ◦A forA∈gl(E), φ∈E∗
and the complex structure of E∗ is given exactly in this way J′∗φ=φ◦J′.
In order to make E∗ a leftgl(E)-module we have to change sign A·φ:=−φ◦A
which with regard to the complex structure corresponds to complex conju-gation.
In fact, integrable modules do not behave well under standard linear algebra operation like the tensor product. The reason is simply that we have to work overCif we want to keep the complex structure on our vector spaces and over R if we want to preserve the g action, since this action is notClinear in general.